Quasi-nilpotency of Generalized Volterra Operators on Sequence Spaces

We study the quasi-nilpotency of generalized Volterra operators on spaces of power series with Taylor coefficients in weighted ℓp spaces 1 < p< + ∞. Our main result is that when an analytic symbol g is a multiplier for a weighted ℓp space, then the corresponding generalized Volterra operator Tg is bounded on the same space and quasi-nilpotent, i.e. its spectrum is { 0 }. This improves a previous result of A. Limani and B. Malman in the case of sequence spaces. Also combined with known results about multipliers of ℓp spaces we give non trivial examples of bounded quasi-nilpotent generalized Volterra operators on ℓp. We approach the problem by introducing what we call Schur multipliers for lower triangular matrices and we construct a family of Schur multipliers for lower triangular matrices on ℓp, 1 < p< ∞ related to summability kernels. To demonstrate the power of our results we also find a new class of Schur multipliers for Hankel operators on ℓ2, extending a result of E. Ricard.